Unit 5. Integration techniques


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1 18.01 EXERCISES Unit 5. Integrtion techniques 5A. Inverse trigonometric functions; Hyperbolic functions 5A1 Evlute ) tn 1 3 b) sin 1 ( 3/) c) If θ = tn 1 5, then evlute sin θ, cos θ, cot θ, csc θ, nd sec θ. d) sin 1 cos(π/6) e) tn 1 tn(π/3) f) tn 1 tn(π/3) g) lim tn 1 x. x b 1 5A Clculte ) b) c). 1 x + 1 b x + b 1 1 x 5A3 Clculte the derivtive with respect to x of the following ( ) ) sin 1 x 1 x + 1 b) tnh x c) ln(x + x + 1) d) y such tht cos y = x, 0 x 1 nd 0 y π/. e) sin 1 (x/) f) sin 1 (/x) g) tn 1 (x/ 1 x ) h) sin 1 1 x 5A4 ) If the tngent line to y = cosh x t x = goes through the origin, wht eqution must stisfy? b) Solve for using Newton s method. 5A5 ) Sketch the grph of y = sinh x, by finding its criticl points, points of inflection, symmetries, nd limits s x nd. COPYRIGHT DAVID JERISON AND MIT 1996, 003 1
2 E Exercises 5. Integrtion techniques b) Give suitble definition for sinh 1 x, nd sketch its grph, indicting the domin of definition. (The inverse hyperbolic sine.) c) Find d sinh 1 x. d) Use your work to evlute + x 5A6 ) Find the verge vlue of y with respect to rclength on the semicircle x + y = 1, y > 0, using polr coordintes. b) A weighted verge of function is / b b f(x)w(x) w(x) Do prt () over gin expressing rclength s ds = w(x). The chnge of vribles needed to evlute the numertor nd denomintor will bring bck prt (). c) Find the verge height of 1 x on 1 < x < 1 with respect to. Notice tht this differs from prt (b) in both numertor nd denomintor. 5B. Integrtion by direct substitution Evlute the following integrls ln x 5B1. x x 1 5B. e 8x 5B3. x cos x 5B4. 5B5. sin x cos x 5B6. sin 7x + 3 sin x 6x 5B7. 5B8. tn 4x 5B9. e x (1 + e x ) 1/3 x + 4 5B10. sec 9x 5B11. sec 9x 5B1. xe x x 5B13.. Hint: Try u = x x 6 Evlute the following integrls by substitution nd chnging the limits of integrtion.
3 5. Integrtion techniques E Exercises π/3 e (ln x) 3/ 1 tn 1 x 5B14. sin 3 x cos x 5B15. 5B x x 5C. Trigonometric integrls Evlute the following 5C1. sin x 5C. sin 3 (x/) 5C3. sin 4 x 5C4. cos 3 (3x) 5C5. sin 3 x cos x 5C6. sec 4 x 5C7. sin (4x) cos (4x) 5C8. tn (x) cos(x) 5C9. sin 3 x sec x 5C10. (tn x + cot x) 5C11. sin x cos(x) (Use double ngle formul.) π 5C1. sin x cos(x) (See 7.) 0 5C13. Find the length of the curve y = ln sin x for π/4 x π/. 5C14. Find the volume of one hump of y = sin x revolved round the xxis. 5D. Integrtion by inverse substitution Evlute the following integrls x 3 (x + 1) 5D1. 5D. 5D3. ( x ) 3/ x 4 + x x 5D4. + x 5D5. 5D6. x + x x (For 5D4,6 use x = sinh y, nd cosh y = (cosh(y) + 1)/, sinh y = sinh y cosh y.) x 5D7. 5D8. x x 9 x 3
4 E Exercises 5. Integrtion techniques 5D9. Find the rclength of y = ln x for 1 x b. Completing the squre Clculte the following integrls 5D10. 5D11. x 8 + 6x x (x + 4x + 13) 5D x x 3/ x 4x 4x D13. 5D14. 5D15. x x x + 4x + 13 x 1 5E. Integrtion by prtil frctions x x 5E1. 5E. 5E3. (x )(x + 3) (x )(x + 3) (x 4)(x + 3) 3x + 4x 11 3x + x 9 5E4. 5E5. 5E6. (x 1)(x ) x(x + 1) (x + 9)(x + ) 5E7 The equlity (1) of Notes F is vlid for x = 1,. Therefore, the equlity (4) is lso vlid only when x = 1,, since it rises from (1) by multipliction. Why then is it legitimte to substitute x = 1 into (4)? 5E8 Express the following s sum of polynomil nd proper rtionl function 3 x x x ) b) c) x 1 x 1 3x 1 x + x 8 d) e) (just give the form of the solution) 3x 1 (x + ) (x ) 5E9 Integrte the functions in Problem 5E8. 5E10 Evlute the following integrls (x + 1) (x + x + 1) ) b) c) x 3 x (x )(x 3) x + 8x 4
5 5. Integrtion techniques E Exercises (x + x + 1) (x + 1) d) e) f) x + 8x x 3 + x x 3 + x + x x 3 (x + 1) g) h) (x + 1) (x 1) x + x + 5E11 Solve the differentil eqution dy/ = y(1 y). 5E1 This problem shows how to integrte ny rtionl function of sin θ nd cos θ using the substitution z = tn(θ/). The integrnd is trnsformed into rtionl function of z, which cn be integrted using the method of prtil frctions. ) Show tht 1 z z dz cos θ =, sin θ =, dθ =. 1 + z 1 + z 1 + z Clculte the following integrls using the substitution z = tn(θ/) of prt (). wy!) π π dθ dθ π b) c) d) sin θdθ (Not the esiest 1 + sin θ (1 + sin θ) E13 ) Use the polr coordinte formul for re to compute the re of the region 0 < r < 1/(1 + cos θ), 0 θ π/. Hint: Problem 1 shows how the substitution z = tn(θ/) llows you to integrte ny rtionl function of trigonometric function. b) Compute this sme re using rectngulr coordintes nd compre your nswers. 5F. Integrtion by prts. Reduction formuls Evlute the following integrls 5F1 ) x ln x ( = 1) b) Evlute the cse = 1 by substitution. 5F ) xe x b) x e x c) x 3 e x 5
6 E Exercises 5. Integrtion techniques d) Derive the reduction formul expressing n x e x in terms of x n 1 e x. 5F3 Evlute sin 1 (4x) 5F4 Evlute e x cos x. (Integrte by prts twice.) 5F5 Evlute cos(ln x). (Integrte by prts twice.) 5F6 Show the substitution t = e x trnsforms the integrl x n e x, into (ln t) n dt. Use reduction procedure to evlute this integrl. 6
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